Although star-forming galaxies span millionfold ranges in their present SFRs
and physical conditions, there is a remarkable continuity in some of
their properties, and these relationships provide
important insights into the physical regulation of the SFR over this
entire spectrum of activities.

We have already seen evidence from
Figures 5 and
7 that the global
SFRs of disks and nuclear starbursts are correlated
with the local gas density, though over very different ranges in
density and SFR per unit area. The left panel
of Figure 9 shows both sets
of data plotted on a common scale, and it reveals that the entire range
of activities, spanning 5-6 orders of magnitude in gas and SFR
densities,
fits on a common power law with index N ~ 1.4
(Kennicutt 1998).
The SFRs for the two sets of data were derived using
separate methods (H
luminosities for the normal disks and FIR
luminosities for the starbursts), and to verify that they are measured
on a self-consistent scale, Figure 9 also shows
H-derived SFR
gas densities for the centers (1-2 kpc) of the normal disks (plotted
as open circles).
The tight relation shows that a simple
Schmidt (1959)
power law provides an excellent empirical parametrization of the SFR,
across an enormous range of SFRs, and it suggests that the gas density
is the primary determinant of the SFR on these scales.

Figure 9. (Left) The global Schmidt
law in galaxies. Solid points denote
the normal spirals in Figure 5,
squares denote the circumnuclear starbursts
in Figure 7. The open circles show the SFRs
and gas densities of the central regions of the normal
disks. (Right) The same SFR data but plotted
against the ratio of the gas density to the average orbital time in the
disk. Both plots are adapted from
Kennicutt (1998).

The uncertainty in the slope of the best-fitting Schmidt law is dominated
by systematic errors in the SFRs, with the largest being the FIR-derived
SFRs and CO-derived gas densities
in the starburst galaxies. Changing either scale individually by a
factor of two introduces a change of 0.1 in the fitted value of N,
and this is a reasonable estimate of the systematic errors involved
(Kennicutt 1998).
Incorporating these uncertainties yields the
following relation for the best-fitting Schmidt law:

(7)

where SFR
and gas
are the disk-averaged SFR and gas surface densities, respectively.

As discussed by
Larson (1992),
Elmegreen (1994),
a large-scale Schmidt law with index N ~ 1.5 would be expected
for self-gravitating disks if
the SFR scales as the ratio of the gas density
() to the
free-fall time
scale (-0.5)
and the average gas scale height is roughly constant across the sample
(). In a variant
on this picture,
Elmegreen (1997),
Silk (1997)
have suggested that the SFR might scale with the ratio of the
gas density to the average orbital time scale; this is equivalent to
postulating that disks process a fixed fraction of their gas into
stars in each orbit around the galactic center. The
right panel of Figure 9, also adapted from
Kennicutt (1998),
shows the correlation between the SFR density and
gas
/ dyn for the same
galaxies and starbursts. For this purpose
dyn is defined as
one disk orbit time, measured at half of the outer
radius of the star-forming disk, in units of years (see
Kennicutt 1998
for details). The line is a median fit to the normal
disks with slope contrained to unity, as predicted by the simple
Silk model. This alternative "recipe" for the SFR provides a fit
that is nearly as good as the Schmidt law. The equation of the fit is

(8)

In this parametrization, the SFR is simply ~ 10% of the available
gas mass per orbit.

These parametrizations offer two distinct interpretations of the high
SFRs in the centers of luminous
starburst galaxies. In the context of the Schmidt law picture,
the star formation efficiency scales as
g(N - 1), or
g0.4 for the observed
relation in Figure 9.
The central starbursts have densities that are on the order of 100-10,000
times higher than in the extended star-forming disks of spirals,
so the global star formation efficiencies should be 6-40 times higher.
Alternatively, in the kinematic picture, the higher efficiencies in the
circumnuclear starbursts are simply a consequence of the shorter
orbital time scales in the galaxy centers, independent of the
gas density. Whether the observed Schmidt law is a consequence
of the shorter dynamical times or vice versa cannot be ascertained
from these data alone, but either description provides an excellent
empirical description or "recipe" for the observed SFRs.

These simple pictures can account for the high SFRs in
the starburst galaxies, as well as for the observed radial variation
of SFRs within star-forming disks
(Kennicutt 1989,
1998).
However the relatively shallow N ~ 1.4 Schmidt law cannot
account for the strong changes in disk SFRs observed across the Hubble
sequence if the disks evolved as nearly closed systems
(Kennicutt et al
1994).
Likewise, the modest changes in galaxy rotation curves with Hubble type
are too small to account for the large differences in star formation
histories with a kinematical model such as in Equation 8. The explanation
probably involves star formation thresholds in the gas-poor galaxies
(Kennicutt 1989,
1997),
but the scenario has not been tested quantitatively, and it is possible
that other mechanisms, such as infall of gas, merger history, or
bulge-disk interactions are responsible for the strong changes in
star formation histories across the spiral sequence.